balls and urn

Consider an urn holding N balls which are either white or black. Whenever I take out a ball it is replaced randomly with either a white or black ball,
with fixed probability p for the replacement ball being white.
I am playing this game for quite a while (so that the start configuration no longer matters) and now I am taking out n white balls in a row (n < N).

What is the probability that the next ball is also white?

If p is (near) 1/2 then one could make these two arguments:

1) If we take out a sequence of n white balls it indicates that there are probably
more white balls in the urn than black balls (due to a random fluctuation in the replacement process), so the next ball is most likely also white: P(white) > P(black).

2) If we take out n white balls, the ratio of white to black necessarily decreases,
so it is more likely that the next is actually black: P(black) > P(white).

What do you think? And does it make a difference if we actually know p and N ?

added later: I have posted the solution now as a comment, but I have to warn you that this is fun only if you really try to find an answer yourself first.


Anonymous said...

I go for 2)

Initially the best guess is that #black = #white = N/2

Now we take out e.g. 10 white and the best guess is that they will be replaced with 5 white and 5 black.

So now the best estimate is #black = N/2 + 5 while #white = N/2 - 10 + 5

Am I right?

wolfgang said...

>> Am I right?
I'll post my answer later.

wolfgang said...

solution: The probability is p and independent of N and n.